reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;

theorem Th3:
  for a being Object of A holds 1Cat(a,id a) is Subcategory of A
proof
  let d be Object of A;
  thus the carrier of 1Cat(d,id d) c= the carrier of A
  proof
    let b be object;
    assume b in the carrier of 1Cat(d,id d);
    then b = d by TARSKI:def 1;
    hence thesis;
  end;
  thus for a,b being Object of 1Cat(d,id d), a9,b9 being Object of A st a = a9
  & b = b9 holds Hom(a,b) c= Hom(a9,b9)
  proof
    let a,b be Object of 1Cat(d,id d), a9,b9 be Object of A;
    assume that
A1: a = a9 and
A2: b = b9;
A3: b9 = d by A2,TARSKI:def 1;
    let x be object;
    assume x in Hom(a,b);
    then
A4: x = id d by TARSKI:def 1;
    a9 = d by A1,TARSKI:def 1;
    hence thesis by A3,A4,CAT_1:27;
  end;
  thus the Comp of 1Cat(d,id d) c= the Comp of A
  proof
    let x be object;
    assume x in the Comp of 1Cat(d,id d);
    then x in {[[id d,id d],id d]} by Th2;
    then x = [[id d,id d],id d] by TARSKI:def 1;
    hence thesis by Th1;
  end;
  let a be Object of 1Cat(d,id d), a9 be Object of A;
  assume a = a9;
  then a9 = d by TARSKI:def 1;
  hence thesis by TARSKI:def 1;
end;
